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    "### 行列式定义\n",
    "\n",
    "对于N阶方阵$A=(a_{ij})_{n\\times n}$，其行列式定义为：\n",
    "$det(A)=\\sum(-1)^ta_{1p1}a_{2p2}\\cdots a_{npn}$，其中t定义为$p1p2\\cdots pn的逆序数$，$\\sum$表示对$12\\cdots n$的全排列组合进行求和。\n",
    "\n",
    "\n",
    "### 向量积(外积或叉积)\n",
    "三维空间的向量$b=\\left(\\begin{split}b1\\\\b2\\\\b3\\end{split}\\right)$,$c=\\left(\\begin{split}c1\\\\c2\\\\c3\\end{split}\\right)$\n",
    "其向量积$S=b\\times c=\\left|\\begin{split}e1\\quad&b1\\quad&c1\\\\e2\\quad&b2\\quad&c2\\\\e3\\quad&b3\\quad&c3\\end{split}\\right|$，其中\n",
    "$e1,e2,e3$是自然基\n",
    "\n",
    "\n",
    "### 2阶行列式与3阶行列式的几何意义\n",
    "对于2阶方阵A，其列向量围成的有向面积被定义为2阶行列式。同理，对于3阶方阵A，其列向量围成的有向体积被定义为3阶行列式。\n",
    "\n",
    "\n",
    "### 余子式与代数余子式\n",
    "在n阶行列式中，把$a_{ij}$所在的第i行和第j列划去后，留下来的n-1阶行列式叫做$a_{ij}$的余子式，记做$M_{ij}$.\n",
    "在$a_{ij}$的余子式$M_{ij}$的基础上，还可以定义$A_{ij}$，称为$a_{ij}$的代数余子式：$A_{ij}=(-1)^{i+j}M_{ij}$\n",
    "\n",
    "### 拉普拉斯展开\n",
    "n阶方阵A=(a_{ij})的行列式可以表示成关于该方阵A的某一行(列)的各元素与其对应的代数余子式乘积之和，即:\n",
    "<br>$det(A)=a_{i1}A_{i1}+\\cdots+a_{in}A_{in}$或者\n",
    "<br>$det(A)=a_{1j}A_{1j}+\\cdots+a_{nj}A_{nj}$\n",
    "\n",
    "### 行列式的性质\n",
    "* $det(A)=det(A^T)$\n",
    "* $det(kA)=k^ndet(A)$\n",
    "* 行（列）互换行列式变号\n",
    "* 某一行（列）的每个元素是两数之和，则此行列式可拆分为两个相加的行列式\n",
    "* $det(AB)=det(A)det(B)$\n",
    "* 三角阵行列式为对角线上元素的乘法\n",
    "* 对角分块矩阵$A=\\left(\\begin{split}B\\quad&O\\newline C\\quad&D\\end{split}\\right)$的行列式$det(A)=det(B)det(C)$\n",
    "\n",
    "### 范德蒙行列式\n",
    "$$det(A)=\\left|\\begin{split}1\\quad&1\\quad&\\cdots\\quad&n\\\\x_1\\quad&x_2\\quad&\\cdots\\quad&x_n\\\\x_1^2\\quad&x_2^2\\quad&\\cdots\\quad&x_n^2\\\\\\vdots\\quad&\\vdots\\quad&\\ddots&\\quad\\vdots\\\\x_1^{n-1}\\quad&x_2^{n-1}\\quad&\\cdots\\quad&x_n^{n-1}\\end{split}\\right|=\\prod_{1\\le j\\lt i\\le n}(x_i-x_j)$$\n",
    "\n",
    "### 伴随矩阵及逆矩阵\n",
    "矩阵A的伴随矩阵$A^*=\\left(\\begin{split}A_{11}\\quad&A_{21}\\quad&\\cdots\\quad&A_{n1}\\\\A_{12}\\quad&A_{22}\\quad&\\cdots\\quad&A_{n2}\\\\\\vdots\\quad&\\vdots\\quad&\\ddots&\\quad\\vdots\\\\A_{1n}\\quad&A_{2n}\\quad&\\cdots\\quad&A_{nn}\\end{split}\\right)$\n",
    "\n",
    "### 逆矩阵\n",
    "对于满秩的方阵A，其逆矩阵$A^{-1}=\\frac{1}{|A|}A^*$"
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